The proximal normal formula in Banach space

Borwein, J. M.; Giles, J. R.

doi:10.1090/S0002-9947-1987-0887515-5

The proximal normal formula in Banach space
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by J. M. Borwein and J. R. Giles PDF

Trans. Amer. Math. Soc. 302 (1987), 371-381 Request permission

Abstract:

Approximation by proximal normals to the Clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis. We give simple general versions of such formulae in infinite dimensional Banach spaces which satisfy different geometrical conditions. Our first class, of spaces with uniformly Gâteaux differentiable norm includes the Hilbert space case and the formulae is attained through dense subsets. Our second class, of reflexive Kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space. Our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of Ekeland and Lebourg. Resume. L’approximation par les normales proximales au sous-différentiel généralisé de Clarke pour une fonction de distance produit d’un ensemble non-vide fermé et le cône normal à l’ensemble produit des normales proximales sont objets d’importance pour l’analyse non-régulière. Nous donnons deux versions simples et générales de telles formules dans les espaces de Banach de dimensions infinies. Premièrement, nous examinons la classe des espaces avec norme uniformément Gâteaux-dérivable qui comprend les espaces de Hilbert. Deuxièmement, nous examinons la classe espaces réflexifs Kadec et lisses. Enfin, notre méthode produit l’existence des solutions pour une classe de problèmes à l’optimisation.

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